Research Journal of Applied Sciences, Engineering and Technology 5(4): 1154-1159, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: July 09, 2012
Accepted: August 08, 2012
Published: February 01, 2013
A Single-Producer Multi-Retailer Integrated Inventory System with Scrap in Production
1
Singa Wang Chiu, 2Kuo-Tsun Liu, 2Chien-Hua Lee and 2Yuan-Shyi Peter Chiu
1
Department of Business Administration,
2
Department of Industrial Engineering and Management, Chaoyang University
of Technology, Taichung 413, Taiwan
Abstract: This study determines the replenishment-distribution policy for a single-producer multi-retailer integrated
inventory system with scrap rate in production. The objective is to find the optimal lot-size and number of shipments
that minimizes the total expected costs for such a specific system. It is assumed that an item is manufactured by a
producer and then delivered to its n different retailers for sale. Each retailer has its own annual product demand and
the demand will be satisfied by synchronized multiple shipments during each production cycle. Unlike the classic
Finite Production Rate (FPR) model assuming perfect production and a continuous inventory issuing policies, the
proposed system assumes that there is an inevitable random defective rate in production and all nonconforming
items are scrap and delivery of product is under a practical multiple shipment policy. Mathematical modeling is
employed and the renewal reward theorem is used to cope with the variable cycle length. Then the expected system
cost function is derived and the convexity of this function is proved. Finally, a closed-form optimal replenishmentdistribution policy for such a specific single-producer multi-retailer integrated system is obtained. A numerical
example is provided to demonstrate the practical usage of the obtained results.
Keywords: FPR model, inventory, manufacturing, multiple retailers, multiple shipments, optimal batch size, scrap
INTRODUCTION
The purpose of this study is to determine the
optimal lot-size and number of shipments that
minimizes the total expected cost for a single-producer
multi-retailer integrated inventory system with scrap in
production. The classic FPR model (also known as the
Economic Production Quantity (EPQ) model) assumes
a perfect production (Taft, 1918; Nahmias, 2009).
However, in a real life manufacturing environment, due
to different unpredictable factors it seems to be
inevitable to have imperfect production (Hadley and
Whitin, 1963; Schneider, 1979; Bielecki and Kumar,
1988; Chern and Yang, 1999; Boone et al., 2000;
Teunter and Flapper, 2003; Ojha et al., 2007; Lin et al.,
2008; Chiu et al., 2009a, 2012a; Lee et al., 2011; Chen,
2011).
Another unrealistic assumption in the classic FPR
model is the continuous inventory issuing policy to
satisfy product demand. In the real world vendor-buyer
integrated system, however, the multiple or periodic
deliveries of finished products are often used. Schwarz
(1973) first examined a one-warehouse N-retailer
deterministic inventory system with the objective of
determining the stocking policy that minimizes the
long-run average system cost per unit time. The optimal
solutions along with a few necessary properties are
derived for such a one-retailer and N identical retailer
problems. Heuristic solutions for the general problem
were also suggested. Goyal (1977) studied the
integrated inventory model for the single suppliersingle customer problem. He proposed a method that is
typically applicable to those inventory problems where
a product is procured by a single customer from a single
supplier and an example was provided to illustrate his
proposed method. Schwarz et al. (1985) considered the
system fill-rate of a one-warehouse N-identical retailer
distribution system as a function of warehouse and
retailer safety stock. They used an approximation model
from a prior study to maximize system fill-rate subject
to a constraint on system safety stock. As results,
properties of fill-rate policy lines are suggested. They
may be used to provide managerial insight into system
optimization and as the basis for heuristics. Banerjee
(1986) studied a joint economic lot-size model for
purchaser and vendor, with the focus on minimizing the
joint total relevant cost. He concluded that a jointly
optimal ordering policy, together with an appropriate
price adjustment, could be economically beneficial for
both parties, but definitely not disadvantageous to either
party. Studies have since been carried out to address
various aspects of vendor-buyer supply chain
optimization issues (Kim and Hwang, 1988; Banerjee
and Burton, 1994; Parija and Sarker, 1999; Cetinkaya
and Lee, 2000; Yao and Chiou, 2004; Hoque, 2008;
Sarker and Diponegoro, 2009; Chiu et al., 2011,
Corresponding Author: Yuan-Shyi Peter Chiu, Department of Industrial Engineering and Management, Chaoyang University
of Technology, Taichung 413, Taiwan
1154
Res. J. Appl. Sci. Eng. Technol., 5(4): 1154-1159, 2013
n
Fig. 1: On-hand inventory of perfect quality items in the
producer side
2012b). Because little attention has been paid to the
area of deriving the joint replenishment-distribution
policy for a single-producer multi-retailer FPR model
with scrap in production, this study intends to bridge
the gap.
MODELING AND ANALYSIS
This study considers a single-producer N-retailer
integrated inventory system with scrap. It is assumed
that the manufacturing process in producer’s side may
randomly produce an x portion of defective items at a
production rate d. All items produced are screened and
unit inspection cost is included in unit production cost
C. All nonconforming items are assumed to be scrap;
they are discarded at the end of production. Under the
normal operation, to prevent shortages from occurring,
the constant production rate P satisfies (P-d-λ)>0 or (1x-λ/P)>0 and d can be expressed as d = Px. Unlike the
classic FPR model assuming a continuous issuing
policy for meeting demand, this research considers a
multi-shipment policy. It is also assumed that the
finished items can only be delivered to retailers if the
whole lot is quality assured at the end of production.
Each retailer has its own annual demand rate λi.
Fixed quantity n installments of the finished batch are
delivered to multiple retailers synchronously at a fixed
interval of time during the downtime t2 (Fig. 1). Cost
parameters used in this study are as follows: the unit
production cost C, unit disposal cost CS, the production
setup cost K, unit holding cost h, the fixed delivery cost
K1i per shipment delivered to retailer i, unit holding cost
h2i for item kept by retailer i and unit shipping cost Ci
for item shipped to retailer i. Additional notation is
listed as follows:
Q
=
= Number of fixed quantity installments of the
finished batch to be delivered to retailers for
each cycle, a decision variable (to be
determined)
m
= Number of retailers
= The production uptime for the proposed
t1
system
H
= Maximum level of on-hand inventory in
units when regular production process ends
t2
= Time required for delivering all quality
assured finished products to retailers
tn
= A fixed interval of time between each
installment of finished products delivered
during production downtime t2
T
= Production cycle length
I (t)
= On-hand inventory of perfect quality items at
time t
Ic (t)
= On-hand inventory at the retailers at
time t
TC (Q, n)
= Total production-inventory-delivery
costs per cycle for the proposed
system
E [TCU (Q, n)] = Total expected production-inventorydelivery costs per unit time for the
proposed system
From Fig. 1, the following equations can be
directly obtained:
T  t1  t2
t1 
T
(1)
Q
H

P Pd
Q

(2)
1  x 
(3)
 1  x  1 
t 2  nt n  Q 
 
P
 
H   P  d  t1   P  d 
m
   i
Q
 1  x  Q
P
(4)
(5)
(6)
i 1
The on-hand inventory of scrap items during
production uptime t1 is:
dt1  Pxt1  xQ.
(7)
Cost for each delivery is:
Production lot size per cycle, a decision
variable (to be determined)
1155 m
K
i 1
1i
1 m

   Ci  iT 
n  i 1

(8)
Res. J. Appl. Sci. Eng. Technol., 5(4): 1154-1159, 2013
Total production-inventory-delivery cost per cycle
TC (Q, n) consists of the setup cost, variable production
cost, disposal cost, fixed and variable delivery cost,
holding cost during production uptime t1 and holding
cost for finished goods kept by both the manufacturer
and the customer during the delivery time t2. Therefore,
TC (Q, n) is:
m
m
i 1
i 1
TC  Q, n   CQ  K  CS  xQ   n K1i   Ci  iT
 H  dt1
n 1 
 1 m
 Tt2

h
 Tt1 
 t1   
 Ht2    h2i  i 
 2n 
 n

 2
 2 i 1
(12)
Because the scrap rate x during production is
assumed to be a random variable with a known
probability density function. In order to take the
randomness of scrap rate into account, the expected
values of x can be used in the cost analysis. Substituting
all parameters from Eq. (1) to (11) in TC (Q, n) and
with further derivations the expected cost E [TCU (Q,
n)] can be obtained as follows:
m
m

 m
 K  n K1i   i CS E  x   i
i 1
i 1


i 1



Q 1  E  x 
1  E  x 
1  E  x
m
E TCU  Q, n   
E TC  Q, n  
E T 
C  i
i 1
m

hQ  i
 n  1  
i 1
  Ci i 


2
2 P 1  E  x   n  
i 1


m
m
hQ i
i 1


  1  E  x  1 
 m
 
P

   i

  i 1
m
 m

h2i i Q 1  E  x 
 h2i iQ   1   1  
 n  1   i 1
      i 1
 

m
2
 n 
P  n 
2 i


i 1


Fig. 2: On-hand inventory in m retailer sides
DERIVATION OF THE OPTIMAL
SOLUTIONS
Total delivery costs for n shipments in a cycle are:
m
n  K 1i 
i 1
m
C
i 1
i
i
T
(9)
The variable holding costs for finished products
kept by the manufacturer, during the delivery time t2
where n fixed-quantity installments of the finished batch
are delivered to customers at a fixed interval of time, are
as follows (Chiu et al., 2009b):
(10)
Tt
m
i 1
2i
  2 E TCU  Q, n  

Q 2
Q n   2
 E TCU  Q, n  


Qn

 2 E TCU  Q, n   

Qn
  Q  0
 

2
 E TCU  Q, n    n 


n 2

From Eq. (13), one obtains:
The variable holding costs for finished products
kept by the customer during the delivery time t2, are as
follows (Fig. 2 and Appendix for details):
h
Proof of convexity: For the proof of convexity of E
[TCU (Q, n)], one can use the Hessian matrix equations
(Rardin, 1998) and verify whether the following
condition (Eq. (14) holds:
(14)
 n 1 
h
 Ht2
 2n 
1
2
(13)

 i  2  T t1 
 n

m
m

 m
 m
h i
h i
 K  n K1i   i
 n  1   i 1

 i 1 
i 1
i 1




Q
2 P 1  E  x   n   2
Q 2 1  E  x 


m
 m

h2i i 
h2i i 1  E  x 



1
1
 n  1   i 1
      i 1
 

m
 n   2   P   n 
2 i




i 1
E TCU  Q, n  
(11)
1156 

  1  E  x  1 
 m
 
P

   i

  i 1
(15)
Res. J. Appl. Sci. Eng. Technol., 5(4): 1154-1159, 2013
m

 m
2  K  n  K1i   i
 i 1
  3 i 1
Q 1  E  x 
 2 E TCU  Q , n  
Q 2
E TCU  Q , n  
n

m
m
i 1
i 1
 K1i   i
Q 1  E  x 
(16)
n* 


m
  1  E  x  1 
 Q  m
  2    h2 i i  h  i   m
 
P
 2n   i 1
i 1

i
 

i 1
(17)
 2 E TCU  Q , n  
n
2


m
m



1
E
x




Q
1


 
  3    h2 i i  h  i   m
 
n
P
   i 1

i 1

i
 

i 1
(18)
m
E TCU  Q, n  
Qn
m
 K   1


Q 1  E  x   2n
i 1
2
1i
i 1
i


m
  1  E  x  1 
 m
h   h i   m
 
2    2i i
P
  i 1
i 1

i
 

i 1
(19)
Substituting Eq. (16), (18) and (19) in Eq. (14) one
has:
  2 E TCU  Q, n  

Q 2
Q n    2
 E TCU  Q, n  


Qn

m
 2 E TCU  Q, n   

2 K  i
Q
Qn



i 1
 0
 
 2 E TCU  Q, n     n  Q 1  E  x 


n 2

(20)
Equation (20) is resulting positive, because K, λ,
(1-E [x]) and Q are all positive. Hence, E [TCU (Q, n)]
is a strictly convex function for all Q and n different
from zero. Therefore, the convexity of E [TCU (Q, n)]
is proved and there exists a minimum of E [TCU
(Q, n)].
Derivation of the optimal policy: To determine jointly
the replenishment-distribution policy for the proposed
single-producer multi-retailer integrated inventory
system with scrap, one can solve the linear system of
Eq. (15) and (17) by setting these partial derivatives
equal to zero. Further derivations one obtains:
It is noted that although in real-world situation the
number of deliveries takes integer values only, Eq. (22)
results in a real number. In order to find the integer
value of n* that minimizes the expected system cost,
two adjacent integers to n must be examined
respectively (Chiu et al., 2009b). Let n+ denote the
smallest integer greater than or equal to n (derived from
Eq. (22)) and n- denote the largest integer less than or
equal to n. Substitute n+ and n- respectively in Eq. (21),
then applying the resulting (Q, n+) and (Q, n-) in
Eq. (13) respectively. By selecting the one that gives
the minimum long-run average cost as the optimal
replenishment-distribution policy (Q*, n*). A numerical
example demonstrating their practical usages is
provided in next section.
NUMERICAL EXAMPLE
Assume a producer can manufacture a product at
an annual rate of 60,000 units. Annual demands λi of
this item from 5 different retailers are 400, 500, 600,
700 and 800 respectively (total demand is 3000/year).
There is a random scrap rate during production uptime
which follows a uniform distribution over the interval
(0, 0.3). Values for additional parameters are:
h
K
C
CS
K1i
*
= Unit holding cost per item at the producer side,
$20 per item per year
= $35000 per production run
= $100 per item
= $20, disposal cost per scrap item
= The fixed delivery cost per shipment for retailer
i, they are $ 100, 200, 300, 400 and 500,
respectively
$480,000
E [TCU (3122,5) = $460,408]
$460,000
m
E [TCU (Q*, n*)]
Q 
(22)


2  K  n K1i   i
i 1

 i 1
m


2
m
1
 m
  1  E  x 
 
 1  E  x    
K   h2i i  h  i  
m
 P 
i 1
 i 1

i
 

i 1
m


m
m
 h  i  m

 1 
i 1
   h2i i  h  i    1  E  x  
K1i 

i 1
i 1
 i 1
 P 
 P





m
 h i  n  1  

2
 m
 1 
i 1


  h 1  E  x     h2i i  h i    1  E  x   
 n 
i 1
 P
 i 1
 P 

 m



2

h

E
x
1







2i i
  i 1

m


n i
i 1


m
(Q*, n*) = (2961,5)
$440,000
(Q*, n*) = (2815,5)
$420,000
$400,000
$380,000
$360,000
0
0.05
0.10
0.15
0.20
0.25
0.30
x
(21)
and
Fig. 3: Variation of random scrap rate effects on the (Q*, n*)
policy and E [TCU (Q*, n*)]
1157 Res. J. Appl. Sci. Eng. Technol., 5(4): 1154-1159, 2013
h2i = unit holding cost for item kept by retailer i, they
are $75, 70, 65, 60 and 55 per item, respectively
Ci = unit transportation cost for item delivered to
retailer i, they are $0.5, 0.4, 0.3, 0.2 and 0.1,
respectively
From Eq. (21), (22) and (13), one obtains the
optimal number of delivery n* = 5, the optimal
replenishment lot size Q* = 3122 and total expected cost
E [TCU (Q*, n*)] = $ 460,408. It is noted that number
of delivery should practically be an integer number
while Eq. (22) results 5.39, by applying the
aforementioned (Q, n+) and (Q, n-) in Eq. (13)
respectively, one finds the optimal number of delivery
n* = 5. Variation of random scrap rate effects on the
optimal (Q*, n*) policy and on the system cost E [TCU
(Q*, n*)] are depicted in Fig. 3. It is noted that as the
random scrap rate x increases, the value of the system
cost E [TCU (Q*, n*)] increases significantly.
CONCLUDING REMARKS
(A2)
nI  t  
 n  D1  I1  t n n  n  1
h21 

I1  t n  1 1  
2
2
2


nI 2  t1  
 n  D2  I 2  t n n  n  1
 h22 


I 2  tn 
2
2
2


(A3)
 n  Dm  I m  t n n  n  1
nI  t  
   h2m 

I m  tn  m 1 
2
2
2 

m
 n  Di  I i  t n n  n  1
nI  t  
  h2i 

I i  tn  i 1 
2
2
2 
i 1

m

h
2i
i 1
2
 n  Di  I i  t n  n  n  1 I i t n  nI i  t1  
Because n installments (fixed quantity D) of the finished lot are
delivered to customer at a fixed interval of time tn, one has the
following:
nI i
(A4)
Di  i t n  I i
(A5)
i
where, Ii denotes number of left over items for each retailer after
demand during each fixed interval of time tn has been satisfied (Fig. 2).
Eq. (A3), the retailers’ holding cost during t2 becomes:
1 m
2
 h2i  Di  Ii  t2   n  1 Iit2  i  t1  
2 i 1 
1 m
2
  h2i  i tn  I i  t2   i t1  t2  i  t1  

2 i 1 

1 m
  Tt

 h2i i 2  i  t1  T 
2 i 1  n
(A6)
REFERENCES
ACKNOWLEDGMENT
The authors appreciate National Science Council of
Taiwan for supporting this research under grant number:
NSC 99-2410-H-324-007-MY3 (3/3).
APPENDIX
From Fig. 1, n installments (fixed quantity D) of the finished lot
are delivered to customer at a fixed interval of time tn, one has:
H
n
t2
n
From Fig. 2, the computations of retailers’ holding cost during t2
(Eq. (11)) are as follows:
t1 
In real world business environments, it is common
to have a producer supplied a product to its multiple
retailers. In such a specific internal supply chains,
management would like to figure out a best
replenishment-distribution policy in order to minimize
total system costs. This study investigates the optimal
replenishment-distribution policy for such a singleproducer multi-retailer integrated inventory system
with scrap rate in production. Mathematical modeling
and analysis are used. The renewal reward theorem is
employed to cope with the variable cycle length (due
to the randomness of scrap rate in production). The
long-run expected production-inventory-delivery cost
per unit time for the proposed system model is derived
and its convexity is proved. Then, a closed-form
solution of the optimal replenishment-distribution
policy for such a single-producer multi-retailer
integrated system is obtained. Effects of various scrap
rates on the optimal solution are also examined. One
interesting topic for future research will be to
investigate the effect of rework on the optimal
replenishment-distribution policy in such a specific
supply chains.
Di 
tn 
(A1)
Banerjee, A., 1986. A joint economic-lot-size model for
purchaser and vendor. Decision Sci., 17(3):
292-311.
Banerjee, A. and J.S. Burton, 1994. Coordinated vs.
independent inventory replenishment policies for a
vendor and multiple buyers. Int. J. Prod. Econ., 35:
215-222.
Bielecki, T. and P.R. Kumar, 1988. Optimality of zeroinventory policies for unreliable production
facility. Oper. Res., 36: 532-541.
Boone, T., R. Ganeshan, Y. Guo and J.K. Ord, 2000.
The impact of imperfect processes on production
run times. Decis. Sci., 31(4): 773-785.
Cetinkaya, S. and C.Y. Lee, 2000. Stock replenishment
and shipment scheduling for vendor managed
inventory systems. Manage. Sci., 46(2): 217-232.
Chen, C.T., 2011. Dynamic preventive maintenance
strategy for an aging and deteriorating production
system. Exp. Syst. Appl., 38(5): 6287-6293.
1158 Res. J. Appl. Sci. Eng. Technol., 5(4): 1154-1159, 2013
Chern, C.C. and P. Yang, 1999. Determining a
threshold control policy for an imperfect
production system with rework jobs. Nav. Res.
Log., 46(2-3): 273-301.
Chiu, S.W., K.K. Chen and J.C. Yang, 2009a. Optimal
replenishment policy for manufacturing systems
with failure in rework, backlogging and random
breakdown. Math. Comput. Model. Dyn., 15(3):
255-274
Chiu, S.W., H.D. Lin, C.B. Cheng and C.L. Chung,
2009b. Optimal production- shipment decisions for
the finite production rate model with scrap. Int.
J. Eng. Model., 22: 25-34
Chiu, Y.S.P., F.T. Cheng and H.H. Chang, 2010.
Remarks on optimization process of manufacturing
system with stochastic breakdown and rework.
Appl. Math. Lett., 23(10): 1152-1155.
Chiu, Y.S.P., S.C. Liu, C.L. Chiu and H.H. Chang,
2011. Mathematical modelling for determining the
replenishment policy for EMQ model with rework
and multiple shipments. Math. Comput. Model.,
54(9-10): 2165-2174.
Chiu, Y.S.P., K.K. Chen and C.K. Ting, 2012a.
Replenishment run time problem with machine
breakdown and failure in rework. Exp. Syst. Appl.,
39(1): 1291-1297.
Chiu, S.W., Y.S.P. Chiu and J.C. Yang, 2012b.
Combining an alternative multi-delivery policy into
economic production lot size problem with partial
rework. Exp. Syst. Appl., 39(3): 2578-2583.
Goyal, S.K., 1977. Integrated Inventory Model for a
Single Supplier-Single Customer Problem. Int.
J. Prod. Res., 15: 107-111.
Hadley, G. and T.M. Whitin, 1963. Analysis of
Inventory Systems. Prentice-Hall, Englewood
Cliffs, New Jersey.
Hoque, M.A., 2008. Synchronization in the singlemanufacturer multi-buyer integrated inventory
supply chain. Eur. J. Oper. Res., 188(3): 811-825.
Kim, K.H. and H. Hwang, 1988. An incremental
discount pricing schedule with multiple customers
and single price break. Eur. J. Oper. Res., 35(1):
71-79.
Lin, H.D., Y.S.P. Chiu and C.K. Ting, 2008. A note on
optimal replenishment policy for imperfect quality
EMQ model with rework and backlogging.
Comput. Math. Appl., 56(11): 2819-2824.
Lee, T.J., S.W. Chiu and H.H. Chang, 2011. On
improving replenishment lot size of an integrated
manufacturing system with discontinuous issuing
policy and imperfect rework. Am. J. Ind. Bus.
Manage., 1(1): 20-29.
Nahmias, S., 2009. Production and Operations
Analysis. McGraw-Hill Co. Inc., New York.
Ojha, D., B.R. Sarker and P. Biswas, 2007. An optimal
batch size for an imperfect production system with
quality assurance and rework. Int. J. Prod. Res.,
45(14): 3191-3214.
Parija, G.R. and B.R. Sarker, 1999. Operations planning
in a supply chain system with fixed-interval
deliveries of finished goods to multiple customers.
IIE Trans., 31(11): 1075-1082.
Rardin, R.L., 1998. Optimization in Operations
Research. Prentice-Hall, New Jersey, pp: 919,
ISBN: 0023984155.
Sarker, B.R. and A. Diponegoro, 2009. Optimal
production plans and shipment schedules in a
supply- chain system with multiple suppliers and
multiple buyers. Eur. J. Oper. Res., 194(3):
753-773.
Schneider, H., 1979. The service level in inventory
control systems. Eng. Costs Prod. Econ., 4:
341-348.
Schwarz, L.B., 1973. A simple continuous review
deterministic one-warehouse N-retailer inventory
problem. Manage. Sci., 19: 555-566.
Schwarz, L.B., B.L. Deuermeyer and R.D. Badinelli,
1986. Fill-rate optimization in a one-warehouse Nidentical retailer distribution system. Manage. Sci.,
31(4): 488-498.
Taft, E.W., 1918. The most economical production lot.
Iron Age, 101: 1410-1412.
Teunter, R.H. and S.D.P. Flapper, 2003. Lot-sizing for
a single-stage single-product production system
with
rework
of
perishable
production
defectives. OR Spectrum, 25(1): 85-96.
Yao, M.J. and C.C. Chiou, 2004. On a replenishment
coordination model in an integrated supply chain
with one vendor and multiple buyers. Eur. J. Oper.
Res., 159: 406-419.
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